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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Substituting (<a href="" class="xref" data-knowl="./knowl/eq3_26.html" title="Equation 3.7.3">(3.7.3)</a>), (<a href="" class="xref" data-knowl="./knowl/eq3_27.html" title="Equation 3.7.4">(3.7.4)</a>) and (<a href="" class="xref" data-knowl="./knowl/eq3_28.html" title="Equation 3.7.6">(3.7.6)</a>) into (<a href="" class="xref" data-knowl="./knowl/eq3_24.html" title="Equation 3.7.1">(3.7.1)</a>), one has</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_26.html ./knowl/eq3_27.html ./knowl/eq3_28.html ./knowl/eq3_24.html ./knowl/eq3_29.html ./knowl/eq3_30.html">
\begin{equation*}
\begin{aligned}
&amp;u_1^{\prime} y_1^{\prime}+u_2^{\prime} y_2^{\prime}+u_1 y_1^{\prime \prime}+u_2 y_2^{\prime \prime}+p(x) (u_1 y_1^{\prime}+u_2 y_2^{\prime})+q(x) (u_1 y_1+u_2 y_2)=g(x)\\
&amp;\to u_1 \underline{\left[y_1^{\prime \prime}+p(x) y_1^{\prime}+q(x) y_1    \right]}+u_2 \underline{\left[y_2^{\prime \prime}+p(x) y_2^{\prime}+q(x) y_2    \right]}+u_1^{\prime} y_1^{\prime}+u_2^{\prime} y_2^{\prime}=g(x),
\end{aligned}
\end{equation*}
</div>
<p class="continuation">which implies</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_26.html ./knowl/eq3_27.html ./knowl/eq3_28.html ./knowl/eq3_24.html ./knowl/eq3_29.html ./knowl/eq3_30.html">
\begin{equation}
u_1^{\prime} y_1^{\prime}+u_2^{\prime} y_2^{\prime}=g(x).\tag{3.7.7}
\end{equation}
</div>
<p class="continuation">We can regard (<a href="" class="xref" data-knowl="./knowl/eq3_29.html" title="Equation 3.7.5">(3.7.5)</a>) and (<a href="" class="xref" data-knowl="./knowl/eq3_30.html" title="Equation 3.7.7">(3.7.7)</a>) as two linear algebraic equations for <span class="process-math">\(u_1^{\prime}\)</span> and <span class="process-math">\(u_2^{\prime}\)</span> and they are solved as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_26.html ./knowl/eq3_27.html ./knowl/eq3_28.html ./knowl/eq3_24.html ./knowl/eq3_29.html ./knowl/eq3_30.html">
\begin{equation*}
\begin{aligned}
&amp;u_1^{\prime}=-\frac{y_2 g(x) }{W(y_1, y_2)},\quad u_2^{\prime}=\frac{y_1 g(x) }{W(y_1, y_2)},\\
&amp;\to u_1=\int -\frac{y_2 g(x) }{W(y_1, y_2)} \textrm{d} x+k_1,\quad u_2=\int \frac{y_1 g(x) }{W(y_1, y_2)} \textrm{d} x+k_2\\
&amp;\to u_1=\int -\frac{y_2 g(x) }{W(y_1, y_2)} \textrm{d} x,\quad u_2=\int \frac{y_1 g(x) }{W(y_1, y_2)} \textrm{d} x,
\end{aligned}
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(k_1, k_2\)</span> are two arbitrary constants and they are set to be <span class="process-math">\(0\text{.}\)</span></p>
<span class="incontext"><a href="sec3_7.html#p-139" class="internal">in-context</a></span>
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